A Conservative whip is one of the 80 MPs appealing against the Commons auditor's demand to pay back expenses.

Jeremy Wright, who has been ordered to repay nearly £800 he claimed for mobile phone bills, appears to be the first Conservative or Labour frontbencher to say that he is contesting a final demand from Sir Thomas Legg, the former civil servant who has reviewed claims submitted by all MPs.

David Cameron originally said that he would remove the whip from any Tory MP who did not pay up in response to a final demand from Legg. But the Commons authorities subsequently introduced an appeals process for MPs who believed they were being unfairly treated by Legg and Wright said he was not defying Cameron because he was still complying with the Legg "process".

Legg, a retired civil servant, has recommended Wright pay back £769.80 he claimed between 2005 and this year in mobile phone bills under the second homes allowance – called the Additional Costs Allowance (ACA).

Such a claim was clearly banned under ACA by parliament's Green Book rule book, which Wright accepts. Any claim for mobile phone bills for work purposes should have been made under the separate office costs allowance, called Incidental Expenses Provision.

The Green Book rules applicable at the time state in section 3.14.1, under ACA, which expenditure is "not allowable". "Mobile phone rental and charges" is included in the list.

Wright, the Conservative MP for Rugby and Kenilworth, said his claim had been sanctioned by the Commons authorities after he had checked with them about using his mobile phone instead of a landline at his rented London flat – which would have been allowed under ACA – as he did not have one.

Some MPs have publicly accepted they should have been guided by the Green Book's key principle that claims should be "wholly, exclusively and necessarily" for the performance or parliamentary duties, whatever the advice from officials, which is also Legg's view.

In a statement, Wright said: "He [Legg] is right on a strict interpretation of the rules, but common sense says otherwise. Had I done as he says I should have, I would have claimed for connection charges and line rental, in addition to the cost of calls, and the claims in question would have been twice as high.

"I believe that in finding economical ways of claiming legitimate expenses, I have done the right thing and that is why I am appealing. If, at the end of the appeal process, I am still asked to pay this money, I will, of course, do so."

Wright said he was therefore acting in accordance with David Cameron's instruction for Tory MPs to comply with "the process" of Legg's audit, and he would not be in trouble with his party.

He said the appeal conducted by retired high court judge Sir Paul Kennedy would have a wider remit than Legg's review and would examine whether an repayment would be "fair and equitable" in all the circumstances, rather than just whether they were against the letter of the rule book.

He told his local newspaper, the Coventry Telegraph: "The Commons members' estimate committee has brought in Sir Paul Kennedy, who I've asked to consider whether there are circumstances of natural justice in this case, and whether repayment would be fair and equitable in the circumstances.

"Sir Thomas Legg makes it clear that his remit is limited. He is not able, within the terms of his review, to consider whether it would be fair and equitable for a repayment to be made. The instructions from the Conservative party leadership are that we comply with the process of Sir Thomas Legg's inquiry. It was not previously known if there would be an appeals process."

Wright received his final letter from Legg on 7 December, after the MP had replied contesting Legg's initial findings. Legg modified his initial findings that Wright should pay back £829.50 of the mobile phone bills.

Wright, on his website, said he had asked the fees office whether claiming a small proportion of his mobile phone bill – £20 each month – would be acceptable in lieu of claiming for a landline, and he was told it would be.

The appeals process is expected to be completed by 15 January, after which MPs will be asked to vote on the members' estimate committee's recommendation that they pay back the sums requested.

Cameron said last week that he believed only a handful of MPs had a genuine grievance.

He said: "There are one or two cases where there is a genuine, really quite fundamental issue and quite a significant amount of money, where the person believes that they have made claims not just within the letter of the law but also the spirit of the rules too and had what they did cleared on several occasions. It's right to have an appeals process for those very hard cases."